Stable pairs on nodal K3 fibrations

TL;DR

This paper analyzes stable pair invariants on nodal K3 fibrations, connecting them with known formulas for K3 surfaces and exploring their relations with perverse invariants and Donaldson-Thomas invariants in Calabi-Yau threefolds.

Contribution

It provides a new description of stable pair invariants on K3 fibrations with nodal fibers using Kawai-Yoshioka's formula and investigates their relation to perverse and Donaldson-Thomas invariants.

Findings

Stable pair invariants are expressed via Kawai-Yoshioka's formula and Noether-Lefschetz numbers.

Relations between stable pair invariants and perverse stable pair invariants are established.

Wall-crossing techniques relate stable pair invariants to Donaldson-Thomas invariants on Calabi-Yau threefolds.

Abstract

We study Pandharipande-Thomas's stable pair theory on $K3$ fibrations over curves with possibly nodal fibers. We describe stable pair invariants of the fiberwise irreducible curve classes in terms of Kawai-Yoshioka's formula for the Euler characteristics of moduli spaces of stable pairs on $K3$ surfaces and Noether-Lefschetz numbers of the fibration. Moreover, we investigate the relation of these invariants with the perverse (non-commutative) stable pair invariants of the $K3$ fibration. In the case that the $K3$ fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniques, we write the stable pair invariants in terms of the generalized Donaldson-Thomas invariants of 2-dimensional Gieseker semistable sheaves supported on the fibers.